The museum is open Fridays through Tuesdays 11 a.m. to 4 p.m. Free timed-entry passes are required. Review our latest visitor safety guidelines.

Painting - Squares of a 3-4-5 Triangle in Scalene Perspective (Dürer)

Painting - Squares of a 3-4-5 Triangle in Scalene Perspective (Dürer)

<< >>
Usage conditions apply
This painting, while similar in subject to the painting entitled Perspective (Alberti), depicts three planes perpendicular to the canvas. These three planes provide a detailed, three-dimensional view of space through the use of perspective. Three vanishing points are implied (though not shown) in the painting, one in each of the three planes.
The painting shows a 3-4-5 triangle surrounded by squares proportional in number to the square of the side. That is, the horizontal plane contains nine squares, the vertical plane contains sixteen squares, and the oblique plane, which represents the hypotenuse of the 3-4-5 triangle, contains twenty-five squares. This explains the extension of the vertical and oblique planes and reminds the viewer of the Pythagorean theorem.
The title of this painting points to the role of the German artist Albrecht Dürer (1471–1528) in creating ways of representing three-dimensional figures in a plane. Dürer is particularly remembered for a posthumously published treatise on human proportion. In his book entitled The Life and Art of Albrecht Dürer, art historian Erwin Panofsky explains that the work of Dürer with perspective demonstrated that the field was not just an element of painting and architecture, but an important branch of mathematics.
This construction may well have originated with Crockett Johnson. However, he may have been influenced by Figure 1 (p. 604) and Figure 3 (p. 608) in Panofsky’s article on Dürer as a Mathematician in The World of Mathematics, edited by James R. Newman (1956). Johnson did not annotate either of these diagrams. The oil painting was completed in 1965 and is signed: CJ65. It is #8 in his series of mathematical paintings.
Currently not on view
Object Name
date made
Duerer, Albrecht
Johnson, Crockett
Physical Description
wood (frame material)
masonite (substrate material)
overall: 82 cm x 65 cm x.6 cm; 32 5/16 in x 25 9/16 in x 1/4 in
ID Number
catalog number
accession number
Credit Line
Ruth Krauss in memory of Crockett Johnson
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Crockett Johnson
Data Source
National Museum of American History
Nominate this object for photography.   

Our collection database is a work in progress. We may update this record based on further research and review. Learn more about our approach to sharing our collection online.

If you would like to know how you can use content on this page, see the Smithsonian's Terms of Use. If you need to request an image for publication or other use, please visit Rights and Reproductions.


Add a comment about this object